A new high-order accurate approach for modeling of wave propagation and heat transfer in heterogeneous materials
Abstract
A nine-month research program is proposed to develop a new high-order accurate numerical approach for modeling of wave propagation and heat transfer in heterogeneous materials. There are the following difficulties with existing numerical approaches: a) accurate numerical solutions of real world problems with complex geometry of boundaries and interfaces require a prohibitively large computation time; b) weak formulations (e.g., the continuous/discontinuous Galerkin approach) that are often used for the derivation of the discrete equations do not lead to the optimal accuracy of discrete systems; c) meshes generated for complex irregular domains and interfaces often include small cut cells; this leads to a significant decrease in accuracy. A new numerical approach with optimal accuracy will be based on simple Cartesian meshes for irregular domains and interfaces and will significantly exceed the accuracy and reduce computation time compared to those for such popular modern numerical approaches as finite elements, isogeometric elements and other high-order methods. It will resolve the issues listed, and will allow accurate and reliable results for real-world wave propagation and heat transfer problems in heterogeneous materials especially with complex irregular interfaces. The objectives of the proposed study are: 1. To develop a new numerical technique for the Poisson equation (e.g., stationary heat transfer problems) on Cartesian meshes for heterogeneous materials with irregular interfaces that includes simple stencil equations with optimal accuracy. This technique will be based on the minimization of the local truncation error, will not require complicated meshes for irregular inter-faces and will yield the optimal order of accuracy. This is a revolutionary new approach for the derivation of numerical methods with optimal accuracy on unfitted Cartesian meshes for irregular interfaces. Preliminary results for the 2-D Poisson equation with simple interfaces (straight lines) show that at computational costs of quadratic finite elements, the new approach provides the 11th order of accuracy and exceeds the accuracy of quadratic finite elements by 8 orders (!). 2. To develop a new high-order numerical technique for time-dependent wave and heat equations on Cartesian meshes for heterogeneous materials with irregular interfaces that includes simple stencil equations with optimal accuracy. This technique will be based on the minimization of the local truncation error, will not require complicated meshes for irregular domains and will yield the optimal order of accuracy. This is a revolutionary new approach for the derivation of numerical methods with optimal accuracy on Cartesian meshes for irregular interfaces. 3. To solve benchmark problems and to show that at the same accuracy the new approach significantly reduces computation time by a factor of 1000 and more compared with known techniques for wave propagation and heat transfer in heterogeneous materials with irregular inter-faces. Intellectual Merit: A new, fast and high-order accurate numerical technique for wave propagation and heat transfer in heterogeneous materials with irregular interfaces will be developed including new fundamental results related to the optimal accuracy of space-discretization techniques and trivial unfitted Cartesian meshes. The new numerical approach will have a trans-formative impact on the development of new numerical techniques for PDEs, will increase ac-curacy and will significantly reduce computation time for real-world problems for heterogeneous materials that are important for many engineering and military applications. Application of the suggested method to real-world problems that cannot be currently solved due to poor accuracy and a prohibitively large computation time of existing methods, will be considered as a longer-term goal. Still another goal is to combine the new approach for PDEs with machine-learning algorithms in
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 25, 2021
- Source ID
- W911NF2110267
Entities
People
- Alexander Idesman
Organizations
- Army Contracting Command
- Texas Tech University
- United States Army